# Optimal packings of bounded degree trees

@article{Joos2019OptimalPO, title={Optimal packings of bounded degree trees}, author={Felix Joos and Jaehoon Kim and Daniel Kuhn and Deryk Osthus}, journal={Journal of the European Mathematical Society}, year={2019} }

We prove that if $T_1,\dots, T_n$ is a sequence of bounded degree trees so that $T_i$ has $i$ vertices, then $K_n$ has a decomposition into $T_1,\dots, T_n$. This shows that the tree packing conjecture of Gyarfas and Lehel from 1976 holds for all bounded degree trees (in fact, we can allow the first $o(n)$ trees to have arbitrary degrees). Similarly, we show that Ringel's conjecture from 1963 holds for all bounded degree trees. We deduce these results from a more general theorem, which yields…

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## References

SHOWING 1-10 OF 44 REFERENCES

### Decomposing almost complete graphs by random trees

- Mathematics, Computer ScienceElectron. Notes Discret. Math.
- 2014

### Packing minor-closed families of graphs into complete graphs

- MathematicsJ. Comb. Theory, Ser. B
- 2016

### Optimal path and cycle decompositions of dense quasirandom graphs

- MathematicsElectron. Notes Discret. Math.
- 2015

### Hamilton decompositions of regular expanders: a proof of Kelly's conjecture for large tournaments

- MathematicsArXiv
- 2012

### An approximate version of the tree packing conjecture

- MathematicsAPPROX-RANDOM
- 2014

We prove that for any pair of constants ɛ > 0 and Δ and for n sufficiently large, every family of trees of orders at most n, maximum degrees at most Δ, and with at most (n2) edges in total packs into…

### Packing Tree Factors in Random and Pseudo-random Graphs

- Mathematics, Computer ScienceElectron. J. Comb.
- 2014

It is proved that for a xed tree T on t vertices and > 0, the random graph Gn;p, with high probability contains a family of edge-disjoint T -factors covering all but an -fraction of its edges, as long as 4 np log 2 n.

### On the Tree Packing Conjecture

- Mathematics, Computer ScienceSIAM J. Discret. Math.
- 2013

The Gyarfas tree packing conjecture is proved and it is proved that any set of trees such that no tree is a star and T_i has n-i+1 vertices packs into K_n (for n) is large enough.

### A Fast Approximation Algorithm for Computing the Frequencies of Subgraphs in a Given Graph

- Mathematics, Computer ScienceSIAM J. Comput.
- 1995

In this paper we give an algorithm which, given a labeled graph on $n$ vertices and a list of all labeled graphs on $k$ vertices, provides for each graph $H$ of this list an approximation to the…

### Packing Trees into the Complete Graph

- MathematicsCombinatorics, Probability and Computing
- 2002

If T is a tree of order n+1−c′n, c′ [les ] 1/25 (37−8 √21 ) ≈ 0.0135748, such that there exists a vertex x ∈ V(T) and T−x has at least n(1−2c′) isolated vertices, then 2n+1 copies of T may be packed into K2n-1.